Question: Solve the exponential equation for $x$. 2 6 x − 8 ⋅ 64 2 − x = 2 9 x + 4 2\^{ 6x-8}\cdot 64\^{ 2-x}=2\^{ 9x+4} $x=$
The strategy Let's write $64$ in base $2$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $2$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 2 6 x − 8 ⋅ 64 2 − x = 2 6 x − 8 ⋅ ( 2 6 ) 2 − x = 2 6 x − 8 ⋅ 2 12 − 6 x = 2 6 x − 8 + ( 12 − 6 x ) = 2 4 ( 64 = 2 6 ) ( ( a n ) m = a n ⋅ m ) ( a n ⋅ a m = a n + m ) \begin{aligned} 2\^{ 6x-8}\cdot 64\^{ 2-x}&=2\^{ 6x-8}\cdot (2^6)\^{ 2-x}&&&&(64=2^6)\\\\ &=2\^{C{6x-8}}\cdot 2\^{ {12-6x}}&&&&((a^n)^m=a^{n\cdot m}) \\\\ &=2\^{ C{6x-8} \ + \ ({12-6x}) }&&&&(a^n\cdot a^m=a^{n + \normalsize m})\\\\ &=2\^{ 4} \end{aligned} Solving the linear equation We obtain the following equation. 2 4 = 2 9 x + 4 2\^{ 4}=2\^{ 9x+4} Now we can equate the exponents and solve for $x$. $\begin{aligned} 4 &=9x+4\\\\ x &=0\end{aligned}$ The answer The answer is $x=0$. You can check this answer by substituting $\it{x=0}$ in the original equation and evaluating both sides.